Coexisting Behavior and Status Transition of the Hodgkin-Huxley Cardiac Purkinje Fiber Model Under External AC Injection

Abstract

The membrane current I_m of the Hodgkin-Huxley (HH) cardiac Purkinje fiber (CPF) model is usually calculated as direct current (DC). In this paper, a conventional alternating current (AC) is used, namely I_{AC} = textrm{Asin}(2 pi ft), as the impressed injection. Dynamic characters of the model with different AC parameters and various initial conditions are observed through phase plane orbits, waveforms, bifurcation diagrams, and Lyapunov exponent spectra, which reveal multiple coexisting membrane action potential patterns, corresponding to the coexisting attractors of the same or different periods in the phase diagram. Meanwhile, the model undergoes period, quasi-period and local forward or inverse period-doubling bifurcations with changes in amplitude A or frequency f, which further proves the complex nonlinear property of the AC-injected model. In addition, by changing the external current I_m, the sodium-ion and potassium-ion equilibrium potentials, i.e. E_{Na} and E_{K}, respectively, the regularity of the CPF heartbeat frequency is observed. The state transformations of CPF are found between normal, abnormal and sudden cardiac arrest, and the method adjusting from the dangerous state to the normal heartbeat frequency range is investigated. This study may provide a reference for exploring the evolution of nonlinear dynamics in HH CPF model and protecting the health of life and heart.